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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 691721, 10 pages
doi:10.1155/2010/691721
Research Article
Existence of Solutions for Nonlinear Fractional
Integro-Differential Equations with Three-Point
Nonlocal Fractional Boundary Conditions
Ahmed Alsaedi and Bashir Ahmad
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203,
Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Bashir Ahmad, bashir
Received 17 March 2010; Revised 6 May 2010; Accepted 11 June 2010
Academic Editor: Kanishka Perera
Copyright q 2010 A. Alsaedi and B. Ahmad. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We prove the existence and uniqueness of solutions for nonlinear integro-differential equations of
fractional order q ∈ 1, 2 with three-point nonlocal fractional boundary conditions by applying
some standard fixed point theorems.
1. Introduction
Fractional calculus differentiation and integration of arbitrary order is proved to be an
important tool in the modelling of dynamical systems associated with phenomena such
as fractal and chaos. In fact, this branch of calculus has found its applications in various
disciplines of science and engineering such as mechanics, electricity, chemistry, biology,
economics, control theory, signal and image processing, polymer rheology, regular variation
in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics
of complex medium, viscoelasticity and damping, control theory, wave propagation,
percolation, identification, and fitting of experimental data 1–4.
Recently, differential equations of fractional order have been addressed by several
researchers with the sphere of study ranging from the theoretical aspects of existence and
uniqueness of solutions to the analytic and numerical methods for finding solutions. For some
recent work on fractional differential equations, see 5–11 and the references therein.
In this paper, we study the following nonlinear fractional integro-differential
equations with three-point nonlocal fractional boundary conditions
D
q
x
t
f
t, x
t
,
φx
t
,
ψx
t
0, 0 <t<1, 1 <q≤ 2,
D
q−1/2
x
0
0,aD
q−1/2
x
1
x
η
0, 0 <η<1,
1.1
2 Advances in Difference Equations
where D is the standard Riemann-Liouville fractional derivative, f : 0, 1 × X × X × X → X
is continuous, for γ,δ: 0, 1 × 0, 1 → 0, ∞,
φx
t
t
0
γ
t, s
x
s
ds,
ψx
t
t
0
δ
t, s
x
s
ds,
1.2
and a ∈ R satisfies the condition aΓqη
q−1
Γq 1/2
/
0. Here, X, · is a Banach space
and C C0, 1,X denotes the Banach space of all continuous functions from 0, 1 → X
endowed with a topology of uniform convergence with the norm denoted by ·.
We remark that fractional boundary conditions result in the existence of both electric
and magnetic surface currents on the strip and are similar to the impedance boundary
conditions with pure imaginary impedance, and in the physical optics approximation, the
ratio of the surface currents is the same as for the impedance strip. For the comparison
of the physical characteristics of the fractional and impedance strips such as radiation
pattern, monostatic radar cross-section, and surface current densities, see 12. The concept
of nonlocal multipoint boundary conditions is quite important in various physical problems
of applied nature when the controllers at the end points of the interval under consideration
dissipate or add energy according to the censors located at intermediate points. Some recent
results on nonlocal fractional boundary value problems can be found in 13–15.
2. Preliminaries
Let us recall some basic definitions 1–3 on fractional calculus.
Definition 2.1. The Riemann-Liouville fractional integral of order q is defined as
I
q
g
t
1
Γ
q
t
0
g
s
t − s
1−q
ds, q > 0,
2.1
provided the integral exists.
Definition 2.2. The Riemann-Liouville fractional derivative of order q for a function gt is
defined by
D
q
g
t
1
Γ
n − q
d
dt
n
t
0
g
s
t − s
q−n1
ds, n − 1 <q≤ n, q > 0,
2.2
provided the right-hand side is pointwise defined on 0, ∞.
Lemma 2.3 see 16. For q>0, let x, D
q
x ∈ C0, 1 ∩ L0, 1.Then
I
q
D
q
x
t
x
t
c
1
t
q−1
c
2
t
q−2
··· c
n
t
q−n
,
2.3
where c
i
∈ R,i 1, 2, ,n(n is the smallest integer such that n ≥ q).
Advances in Difference Equations 3
Lemma 2.4 see 2. Let x ∈ L0, 1.Then
i D
ν
I
μ
xtI
μ−ν
xt,μ >ν >0;
ii D
μ
t
ξ−1
Γξ/Γξ − μt
ξ−μ−1
,μ>0,ξ >0.
Lemma 2.5. For a given σ ∈ C0, 1 ∩ L0, 1, the unique solution of the boundary value problem
D
q
x
t
σ
t
0, 0 <t<1, 1 <q≤ 2,
D
q−1/2
x
0
0,aD
q−1/2
x
1
x
η
0, 0 <η<1,
2.4
is given by
x
t
−
t
0
t − s
q−1
Γ
q
σ
s
ds
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
η
0
η − s
q−1
Γ
q
σ
s
ds a
1
0
1 − s
q−1/2
Γ
q 1
/2
σ
s
ds
.
2.5
Proof. In view of Lemma 2.3, the fractional differential equation in 2.4 is equivalent to the
integral equation
x
t
−I
q
σ
t
b
1
t
q−1
b
2
t
q−2
−
t
0
t − s
q−1
Γ
q
σ
s
ds b
1
t
q−1
b
2
t
q−2
,
2.6
where b
1
,b
2
∈ R are arbitrary constants. Applying the boundary conditions for 2.4,wefind
that b
2
0and
b
1
Γ
q 1
/2
aΓ
q
η
q−1
Γ
q 1
/2
η
0
η − s
q−1
Γ
q
σ
s
ds a
1
0
1 − s
q−1/2
Γ
q 1
/2
σ
s
ds
.
2.7
Substituting the values of b
1
and b
2
in 2.6,weobtain2.5. This completes the proof.
3. Main Results
To establish the main results, we need the following assumptions.
A
1
There exist positive functions L
1
t,L
2
t,L
3
t such that
f
t, x
t
,
φx
t
,
ψx
t
− f
t, y
t
,
φy
t
,
ψy
t
≤ L
1
t
x − y
L
2
t
φx − φy
L
3
t
ψx − ψy
, ∀t ∈
0, 1
,x,y∈ X.
3.1
4 Advances in Difference Equations
Further,
γ
0
sup
t∈
0,1
t
0
γ
t, s
ds
,δ
0
sup
t∈
0,1
t
0
δ
t, s
ds
,
I
q
L
sup
t∈
0,1
{|
I
q
L
1
t
|
,
|
I
q
L
2
t
|
,
|
I
q
L
3
t
|}
,
I
q1/2
L
1
max
I
q1/2
L
1
1
,
I
q1/2
L
2
1
,
I
q1/2
L
3
1
,
I
q
L
η
max
I
q
L
1
η
,
I
q
L
2
η
,
I
q
L
3
η
.
3.2
A
2
There exists a number κ such that Λ ≤ κ<1, where
Λ
1 γ
0
δ
0
I
q
L
λ
1
I
q
L
η
|
a
|
I
q1/2
L
1
,
λ
1
Γ
q 1
/2
aΓ
q
η
q−1
Γ
q 1
/2
.
3.3
A
3
ft, xt, φxt, ψxt≤μt, for all t, x, φx, ψx ∈ 0, 1 × X × X × X, μ ∈
L
1
0, 1,R
.
Theorem 3.1. Assume that f : 0, 1 ×X ×X ×X → X is a jointly continuous function and satisfies
the assumption A
1
. Then the boundary value problem 1.1 has a unique solution provided Λ < 1,
where Λ is given in the assumption A
2
.
Proof. Define : C → C by
x
t
−
t
0
t − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
ds
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
η
0
η − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
ds
a
1
0
1 − s
q−1/2
Γ
q 1
/2
f
s, x
s
,
φx
s
,
ψx
s
ds
,t∈
0, 1
.
3.4
Let us set sup
t∈0,1
|ft, 0, 0, 0| M, and choose
r ≥
M
1 − λ
1 λ
1
η
q
Γ
q 1
λ
1
|
a
|
Γ
q 3
/2
, 3.5
Advances in Difference Equations 5
where λ is such that Λ ≤ λ<1. Now we show that B
r
⊂ B
r
, where B
r
{x ∈ C : x≤r}.
For x ∈ B
r
, we have
x
t
≤
t
0
t − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
ds
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
η
0
η − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
ds
|
a
|
1
0
1 − s
q−1/2
Γ
q 1
/2
f
s, x
s
,
φx
s
,
ψx
s
ds
≤
t
0
t − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
− f
s, 0, 0, 0
f
s, 0, 0, 0
ds
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
η
0
η − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
− f
s, 0, 0, 0
f
s, 0, 0, 0
ds
|
a
|
1
0
1−s
q−1/2
Γ
q1
/2
f
s, x
s
,
φx
s
,
ψx
s
−f
s, 0, 0, 0
f
s, 0, 0, 0
ds
≤
t
0
t − s
q−1
Γ
q
L
1
s
x
s
L
2
s
φx
s
L
3
s
ψx
s
M
ds
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
η
0
η − s
q−1
Γ
q
L
1
s
x
s
L
2
s
φx
s
L
3
s
ψx
s
M
ds
|
a
|
1
0
1 − s
q−1/2
Γ
q 1
/2
L
1
s
x
s
L
2
s
φx
s
L
3
s
ψx
s
M
ds
≤
t
0
t − s
q−1
Γ
q
L
1
s
x
s
γ
0
L
2
s
x
s
δ
0
L
3
s
x
s
M
ds
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
6 Advances in Difference Equations
×
η
0
η − s
q−1
Γ
q
L
1
s
x
s
γ
0
L
2
s
x
s
δ
0
L
3
s
x
s
M
ds
|
a
|
1
0
1 − s
q−1/2
Γ
q 1
/2
L
1
s
x
s
γ
0
L
2
s
x
s
δ
0
L
3
s
x
s
M
ds
≤
I
q
L
1
t
γ
0
I
q
L
2
t
δ
0
I
q
L
3
t
r
Mt
q
Γ
q 1
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
I
q
L
1
η
γ
0
I
q
L
2
η
δ
0
I
q
L
3
η
r
Mη
q
Γ
q 1
|
a
|
I
q1/2
L
1
1
γ
0
I
q1/2
L
2
1
δ
0
I
q1/2
L
3
1
r
M
Γ
q 3
/2
≤
1 γ
0
δ
0
I
q
L
λ
1
I
q
L
η
|
a
|
I
q1/2
L
1
r M
1 λ
1
η
q
Γ
q 1
λ
1
|
a
|
Γ
q 3
/2
≤
Λ1 − λ
r ≤ r.
3.6
Now, for x, y ∈ C and for each t ∈ 0, 1, we obtain
x
t
−
y
t
≤
t
0
t − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
− f
s, y
s
,
φy
s
,
ψy
s
ds
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
η
0
η − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
− f
s, y
s
,
φy
s
,
ψy
s
ds
|
a
|
1
0
1 − s
q−1/2
Γ
q 1
/2
f
s, x
s
,
φx
s
,
ψx
s
− f
s, y
s
,
φy
s
,
ψy
s
ds
≤
t
0
t − s
q−1
Γ
q
L
1
s
x − y
L
2
s
φx − φy
L
3
s
ψx − ψy
ds
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
η
0
η − s
q−1
Γ
q
L
1
s
x − y
L
2
s
φx − φy
L
3
s
ψx − ψy
ds
|
a
|
1
0
1 − s
q−1/2
Γ
q 1
/2
L
1
s
x − y
L
2
s
φx − φy
L
3
s
ψx − ψy
ds
Advances in Difference Equations 7
≤
I
q
L
1
t
γ
0
I
q
L
2
t
δ
0
I
q
L
3
t
x − y
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
I
q
L
1
η
γ
0
I
q
L
2
η
δ
0
I
q
L
3
η
|
a
|
I
q1/2
L
1
1
γ
0
I
q1/2
L
2
1
δ
0
I
q1/2
L
3
1
x − y
≤
1 γ
0
δ
0
I
q
L
λ
1
I
q
L
η
|
a
|
I
q1/2
L
1
x − y
Λ
x − y
,
3.7
where we have used the assumption A
2
.AsΛ < 1, therefore is a contraction. Thus, the
conclusion of the theorem follows by the contraction mapping principle.
Now, we state Krasnoselskii’s fixed point theorem 17 which is needed to prove the
following result to prove the existence of at least one solution of 1.1.
Theorem 3.2. Let M be a closed convex and nonempty subset of a Banach space X. Let A, B be the
operators such that iAx By ∈ M whenever x, y ∈ M; iiA is compact and continuous; iii B is
a contraction mapping. Then there exists z ∈ M such that z Az Bz.
Theorem 3.3. Let f : 0, 1 ×X ×X ×X → X be jointly continuous, and the assumptions A
1
and
A
3
hold with
Λ
1
λ
1
1 γ
0
δ
0
I
q
L
η
|
a
|
I
q1/2
L
1
< 1. 3.8
Then there exists at least one solution of the boundary value problem 1.1 on 0, 1.
Proof. Let us fix
r ≥
μ
L
1
1 λ
1
η
q
Γ
q 1
λ
1
|
a
|
Γ
q 3
/2
, 3.9
and consider B
r
{x ∈ C : x≤r}. We define the operators Θ
1
and Θ
2
on B
r
as
Θ
1
x
t
−
t
0
t − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
ds,
Θ
2
x
t
Γ
q 1
/2
t
q−1
aΓ
q
η
q−1
Γ
q 1
/2
×
η
0
η − s
q−1
Γ
q
f
s, x
s
,
φx
s
,
ψx
s
ds
a
1
0
1 − s
q−1/2
Γ
q 1
/2
f
s, x
s
,
φx
s
,
ψx
s
ds
.
3.10
8 Advances in Difference Equations
For x, y ∈ B
r
, we find that
Θ
1
x Θ
2
y
≤
μ
L
1
1 λ
1
η
q
Γ
q 1
λ
1
|
a
|
Γ
q 3
/2
≤ r. 3.11
Thus, Θ
1
x Θ
2
y ∈ B
r
. It follows from the assumption A
1
that Θ
2
is a contraction mapping
for Λ
1
< 1.
In order to prove that Θ
1
is compact and continuous, we follow the approach used
in 6, 7.Continuityoff implies that the operator Θ
1
xt is continuous. Also, Θ
1
xt is
uniformly bounded on B
r
as
Θ
1
x
≤
μ
L
1
Γ
q 1
.
3.12
Now, we show that Θ
1
xt is equicontinuous. Since f is bounded on the compact set
0, 1 × B
r
× B
r
× B
r
, therefore, we define sup
t,x,φx,ψx∈0,1×B
r
×B
r
×B
r
ft, x,φx, ψx f
max
.
Consequently, for t
1
,t
2
∈ 0, 1, we have
Θ
1
x
t
1
−
Θ
1
x
t
2
1
Γ
q
t
1
0
t
2
− s
q−1
−
t
1
− s
q−1
f
s, x
s
,φx
s
,ψx
s
ds
t
2
t
1
t
2
− s
q−1
f
s, x
s
,φx
s
,ψx
s
ds
≤
f
max
Γ
q 1
2
t
2
− t
1
q
t
q
1
− t
q
2
,
3.13
which is independent of x. So, Θ
1
is relatively compact on B
r
. Hence, By Arzela-Ascoli’s
Theorem, Θ
1
is compact on B
r
. Thus all the assumptions of Theorem 3.2 are satisfied and the
conclusion of Theorem 3.2 implies that the boundary value problem 1.1 has at least one
solution on 0, 1.
Example. Consider the following boundary value problem:
c
D
3/2
x
t
t
8
|
x
|
1
|
x
|
1
5
t
0
e
−s−t
5
x
s
ds
1
5
t
0
e
−s−t/2
5
x
s
ds 0,t∈
0, 1
,
D
1/4
x
0
0,aD
1/4
x
1
x
1
3
0.
3.14
Advances in Difference Equations 9
Here, q 3/2,γt, se
−s−t
/5,δ e
−s−t/2
/5,a 1,η 1/3. With γ
0
e − 1/5,δ
0
2
√
e − 1/5, we find that
Λ
8
e 2
√
e 1
9
√
3π Γ
1/4
225
√
π
2
√
3π Γ
1/4
< 1. 3.15
Thus, by Theorem 3.1, the boundary value problem 3.14 has a unique solution on 0, 1.
4. Conclusions
This paper studies the existence and uniqueness of solutions for nonlinear integro-differential
equations of fractional order q ∈ 1, 2 with three-point nonlocal fractional boundary
conditions involving the fractional derivative D
q−1/2
x·. Our results are based on a
generalized variant of Lipschitz condition given in A
1
, that is, there exist positive functions
L
1
t,L
2
t, and L
3
t such that
f
t, x
t
,
φx
t
,
ψx
t
− f
t, y
t
,
φy
t
,
ψy
t
≤ L
1
t
x − y
L
2
t
φx − φy
L
3
t
ψx − ψy
, ∀t ∈
0, 1
,x,y∈ X.
4.1
In case L
1
t,L
2
t,andL
3
t are constant functions, that is, L
1
tL
1
,L
2
tL
2
,andL
3
t
L
3
L
1
,L
2
, and L
3
are positive real numbers, then Lipschitz-generalized variant reduces to
the classical Lipschitz condition and Λ in the assumption A
2
takes the form
Λ
L
1
γ
0
L
2
δ
0
L
3
1 λ
1
η
q
Γ
q 1
λ
1
|
a
|
Γ
q 3
/2
. 4.2
In the limit q → 2, our results correspond to a second-order integro-differential equation
with fractional boundary conditions:
D
2
x
t
f
t, x
t
,
φx
t
,
ψx
t
0, 0 <t<1,
D
1/2
x
0
0,aD
1/2
x
1
x
η
0, 0 <η<1.
4.3
Acknowledgment
The authors are grateful to the referees for their careful review of the manuscript.
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